DMS Linear Algebra/Algebra Seminar

Speaker: Wei Gao

Title: Sign patterns that require H_n and its generalization to zero-nonzero patterns

Abstract: The refined inertia of a square real matrix is the ordered 4-tuple (n_+, n_-, n_z, 2n_p), where n_+ (resp., n_-) is the number of eigenvalues with positive (resp., negative) real part, n_z is the number of zero eigenvalues and 2n_p is the number of pure imaginary eigenvalues.

The set of refined inertias H_n=(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0) is important for the onset of Hopf bifurcation in dynamical systems. In this talk, I will introduce some results about sign patterns, i.e., matrices whose entries are from the set {+, -, 0}, that require H_n.

Recently, Berliner et al. expend H_n to H_n* for zero-nonzero patterns, i.e., matrices whose entries are from the set {0, *}. In this talk, I will show that there is no zero-nonzero pattern that requires H_n*.​